author	title
Bharathi Ramana Joshi	Lectures notes for the Mathematical Models in Biology course, IIIT Hyderbad Monsoon 2021

Lecture 1 : 17/08/2021

Model exploits math to represent, analyze or make predictions about real world phenomenon.
Models as "replica" of real-life systems.
Mathematical models as differential equations.
Why need modelling?
1. Infeasible experiments.
2. Complicated results that are difficult to comprehend.
Uses of models
- Gives physical explanation of a phenomenon.
- Helps make predictions.
- Helps build hypothesis.
Biological processes are not static.
Dynamical systems: systems changing with times. Examples:
1. COVID modelling.
2. Number of predators and preys in ecosystem.
3. Infections disease modelling.
4. Gene expansion (transcription, translation).
5. Signal transduction.
6. Growth of bacteria in a fermenter.
7. Sleep cycle.

Differential equations

Order of a differential equation = order of highest derivative.
Degree of a differential equation = power of highest derivative.
Linear differential equation: $$ a_0(x)y^{(n)} + a_1(x)y^{(n - 1)} + \dots + a_n(x)y = F(x) $$

where $a_i(x)$ and $F(x)$ are single variable functions of $x$.

Malthusian model: the rate of growth of population is directly proportional to the population at that moment.
Modify Malthusian model to incorporate more complex features.
Carrying capacity: limit of a population as time tends to infinity.